Washington and Moscow clashed Wednesday over who actually killed a top Islamic State commander, with the Pentagon pushing back against Russia’s claim that a Russian airstrike — not an American one — successfully targeted Abu Muhammad al-Adnani, one of the founding members and a chief strategist of the brutal jihadi movement.

“We have no information to support Russia’s claim that they also carried out a strike against Adnani,” said Pentagon spokesman Peter Cook, questioning Moscow’s contention that the commander was killed by a Russian bomb on Tuesday.

The bulk of Russia’s military campaign in Syria has been spent “supporting and propping up” the regime of Syrian President Bashar Assad, Mr. Cook told reporters at the Pentagon on Wednesday afternoon, adding that Moscow “has not devoted much, if any, effort that we’re aware of targeting ISIL’s leadership.”

“At the same time,” he said, “we have not seen the Russian military campaign use precision weaponry on a regular basis.”

The lack of clarity on the Adnani strike, meanwhile, underscores the chaos that continues to envelop the 5-year-old, multifront civil war in Syria, where the U.S. is far from the lone international power jockeying for influence. The situation has grown only more complex since last week, when the Turkey suddenly opened its largest military incursion into Syria since the start of the war.

Turkey’s involvement in the U.S.-led campaign against the Islamic State has for years caused concern among U.S. officials, wary that Ankara’s real goal is to create a pretext for crushing Kurdish militants whom Washington has relied upon to retake territory in northern Syria and Iraq.

An uneasy truce between Turkish troops and Kurdish-led forces in northern Syria held on Wednesday, despite Ankara’s vow that it would never negotiate with Syrian Kurdish fighters that it deems as terrorists no less dangerous than the Islamic State.

The Obama administration has called on the Turks and the Kurds to stop fighting each other and focus on defeating the Islamic State, hoping to halt days of clashes between a NATO ally and a Kurdish force that has proven to be highly effective against the Islamic State.

But a spokesman for Turkish President Recep Tayyip Erdogan said Turkey would continue to attack U.S.-backed Kurdish militias inside Syria. The spokesman, Ibrahim Kalin, said a cease-fire was “out of the question,” according to The Associated Press.

Turkey views the Syrian Kurdish fighters as an offshoot of the Kurdistan Workers’ Party, or PKK, which has waged a decades-long insurgency in Turkey and is viewed as a terrorist group by Turkey and the U.S.

Turkey’s stepped-up campaign is complicating an already complex negotiation between the U.S. and Russia on a possible cease-fire in the civil war in order to concentrate on the fight against Islamic State and other jihadi groups. After a meeting last week, Russian Foreign Minister Sergey Lavrov and Secretary of State John F. Kerry said they had agreed on key steps for reinstating a cease-fire that held briefly before faltering in April.

State Department spokesman John Kirby told reporters Wednesday that the talks could also lead to some level of military-to-military coordination between Russian and American forces in Syria.

The lack of such coordination has added to confusion over the conflicting claims of credit surrounding the airstrikes carried out against a top Islamic State commander this week.

The Islamic State — also known as ISIS and ISIL — issued a statement of its own on Tuesday, saying that Adnani, the public face of the group’s worldwide jihadi propaganda campaign and the commander responsible for its attacks against Europe and elsewhere, had been killed.

According to a tweet by the Amaq Agency, the terror group’s online propaganda operation, Adnani was killed in northern Syria’s Aleppo province. The tweet only acknowledged that Adnani was killed “while surveying the operations to repel military campaigns” in the northern Syrian province.

Mr. Cook and other Pentagon officials have since said Adnani was the target of a U.S. airstrike near al-Bab in Aleppo, though the Defense Department has so far declined to confirm the commander’s death.